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Full text of "Treatise On Analysis Vol-Ii"

6 LOCALLY FINITE COVERINGS AND PARTITIONS OF UNITY 21 Put Kn = 0W Vn,l9 which is a compact set for all n g: 0 (we put Un = 0 for all n < 0). For each n *> 0, the open set Vn + l 0^-2 is a neighborhood of Kn. Hence, for each x e Kn, there exists an open neighborhood WŁn) 93 of jc, contained in U^+x UM^.2 a°d contained in one of the sets A^. There exists a finite number of points xt e Kn (1 <Ł / <; /?) such that the sets W^} cover Kn. Arrange the sets W^l) (m ^ 0, 1 g / <; /?, for each m) in a sequence in any way, and let (Bn) denote this sequence. It is clear that (Bn) is an open covering of E which is finer than (AA)A 6 L, and that the sets Bn are relatively compact. Thus all that has to be checked is that the covering (Bn) is locally finite. Let z be any point of E, and let n be the least integer such that z e UM. Since z$Un-it there exists a neighborhood T of z contained in Un and not meeting 0M_2. Consequently T can intersect only the sets W^ for which n 2^m^n+ 1, and the number of these sets is finite. (12.6.2) Let (An) be a denumerable locally finite open covering of a metrizable space E. Then there exists an open covering (BJ ofE such that Bn c Anfor alln. We shall define the family (BM) by induction on n, in such a way that Bn c An for each n, and such that for each n the family consisting of the Bfc with k^n and the Ay- withy > /? is an open covering of E. Suppose that the sets Bn have been defined for n < m. Then the Bn with n < m and the Ay with j ^m cover E. Let C be the open set which is the union of the Brt with n < m and the Aj withy > m -f 1, so that we have E Am c C. We shall show that there exists an open set V such that E Am c= V c V c C. If E = Am, we may take V = 0. If C = E, we may take V = E. If neither E Am nor E C is empty, there exists a continuous mapping / of E into the interval [0, 1] of R, which is equal to 0 on E - Am and to 1 on E - C (4.5.2). We then take V to be the open set of points y such that/(j/) < Ł, and then V is con- tained in the closed set of points y such that f(y) ^ ^, and hence V c C. Put Bw = E - V. Then we have Bm <= E - V c Am and B,,, u C = E, so that the sets Bnwith n ^ m satisfy the required conditions, and the induction can proceed. For each x e E, there exists an integer n such that x $ Am for all m > n, and therefore x e Bk for some k ^ n. Hence the sets BM cover E. It is clear that the same argument will apply if the covering (An) is finite. If E is a topological space and/is a mapping of E into a real vector space F or into K, the support off (denoted by Supp(/)) is defined to be the smallest closed set S in E such that/vanishes on E S. In other words, Supp(/) is the closure in E of the set of points x e E such that/(x) ^ 0; or again it is the set of points x e E with the property that every neighborhood of x in E contains a point y such that/0/) ^ 0.a nonempty open interval in R, there exists no nonconstant mapping